Let $$C=\{z\in \mathbb C, {\rm Re}(z^2)=25 \}$$
Is $C$ open or closed?
I am a little confused here. I think the set should be open and closed, because since the imaginary part has not been defined, the set can include the entire complex plane, when ${\rm Re}(z)=-5,5$. Is my reasoning true?
The set $C$ is closed because it is of the form $\{z \in \Bbb C \mid f(z) = g(z) \}$ with $f(z) = {\rm Re}(z^2)$ and $g \equiv 25$ continuous functions and $\Bbb C$ Hausdorff. Since $\Bbb C$ is connected, $C$ can't be open too.
Maybe you'll want to check for yourself that $z \mapsto {\rm Re}(z^2)$ is continuous and that if $X, Y$ are topological spaces, with $Y$ Hausdorff, and $f,g\colon X \to Y$ are continuous, then $\{x \in X \mid f(x) = g(x)\}$ is closed.